I was toying around when I noticed for $a,b > 0$:
$$f(ab) = f(a)f(b) + f(a-1)f(b-1)$$
is satisfied by $f(n) = T_n$ ; the triangular numbers $n(n+1)/2$.
This equation is not an addition formula, not an abel function nor a fibonacci or Somos type recursion. So I liked that.
So I was wondering about similar equations like
$$f_n(ab) = f_n(a)f_n(b) + f_n(a-1)f_n(b-1) + f_n(a-2)f_n(b-2) + ...+f_n(a-n)f_n(b-n)$$
OR
$$f_W(ab) = f_W(a)f_W(b) + f_W(a-1)f_W(b-1) + f_W(a-2)f_W(b-2) + ...+f_W(0)f_W(0)$$
Can they be understood by the binomium of newton ?
Are they related to the rising factorial ?
The equations look familiar.
I assume they are all polynomials.
Any ideas ?